Newform orbit 630.4.bo.a

• Label: 630.4.bo.a
• Level: $630$
• Weight: $4$
• Character orbit: 630.bo
• Analytic conductor: $37.171$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	630.bo (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(37.1712033036\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{2} - 96 q^{4} - 18 q^{5} + 384 q^{8}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
89.1	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−11.1792	+	0.162607i		0		−17.7270	+	5.36235i	8.00000				0		10.8975	−	19.5255i
89.2	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−11.1374	+	0.979228i		0		14.6848	+	11.2852i	8.00000				0		9.44130	−	20.2697i
89.3	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−10.9183	+	2.40634i		0		3.57853	−	18.1712i	8.00000				0		6.75041	−	21.3174i
89.4	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−10.5371	+	3.73749i		0		−15.7564	+	9.73321i	8.00000				0		4.06361	−	21.9883i
89.5	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−10.3654	−	4.19025i		0		−9.44916	−	15.9284i	8.00000				0		17.6231	−	13.7632i
89.6	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−8.81157	−	6.88159i		0		9.44916	+	15.9284i	8.00000				0		20.7308	−	8.38050i
89.7	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−7.29745	+	8.47037i		0		16.1725	+	9.02502i	8.00000				0		−7.37367	−	21.1099i
89.8	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−5.44876	−	9.76274i		0		17.7270	−	5.36235i	8.00000				0		22.3583	+	0.325214i
89.9	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−4.72065	−	10.1349i		0		−14.6848	−	11.2852i	8.00000				0		22.2747	+	1.95846i
89.10	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−3.37521	−	10.6587i		0		−3.57853	+	18.1712i	8.00000				0		21.8366	+	4.81268i
89.11	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−2.15234	+	10.9712i		0		12.3205	−	13.8277i	8.00000				0		−16.8504	−	14.6992i
89.12	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−2.03180	−	10.9942i		0		15.7564	−	9.73321i	8.00000				0		21.0743	+	7.47498i
89.13	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	−1.29121	+	11.1055i		0		−8.88443	−	16.2501i	8.00000				0		−17.9441	−	13.3420i
89.14	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	0.654331	+	11.1612i		0		−15.1516	+	10.6503i	8.00000				0		−19.9861	−	10.0278i
89.15	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	0.914226	+	11.1429i		0		−1.14987	+	18.4845i	8.00000				0		−20.2143	−	9.55941i
89.16	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	3.68683	−	10.5550i		0		−16.1725	−	9.02502i	8.00000				0		14.5949	+	16.9407i
89.17	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	7.17618	+	8.57336i		0		18.5106	+	0.596872i	8.00000				0		−22.0257	+	3.85614i
89.18	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	8.42518	−	7.34959i		0		−12.3205	+	13.8277i	8.00000				0		4.30468	+	21.9424i
89.19	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	8.97207	−	6.67098i		0		8.88443	+	16.2501i	8.00000				0		2.58241	+	22.2111i
89.20	−1.00000	+	1.73205i		0		−2.00000	−	3.46410i	9.30153	+	6.20335i		0		13.6176	+	12.5523i	8.00000				0		−20.0460	+	9.90737i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
15.d	odd	2	1	inner
105.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	630.4.bo.a	✓	48
3.b	odd	2	1		630.4.bo.b	yes	48
5.b	even	2	1		630.4.bo.b	yes	48
7.d	odd	6	1	inner	630.4.bo.a	✓	48
15.d	odd	2	1	inner	630.4.bo.a	✓	48
21.g	even	6	1		630.4.bo.b	yes	48
35.i	odd	6	1		630.4.bo.b	yes	48
105.p	even	6	1	inner	630.4.bo.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.4.bo.a	✓	48	1.a	even	1	1	trivial
630.4.bo.a	✓	48	7.d	odd	6	1	inner
630.4.bo.a	✓	48	15.d	odd	2	1	inner
630.4.bo.a	✓	48	105.p	even	6	1	inner
630.4.bo.b	yes	48	3.b	odd	2	1
630.4.bo.b	yes	48	5.b	even	2	1
630.4.bo.b	yes	48	21.g	even	6	1
630.4.bo.b	yes	48	35.i	odd	6	1